Bài tập bất đẳng thức và bất phương trình

Trung Tâm Luyện Thi Đại Học Amsterdam

Chương IV-Bài 1. Bất Đẳng Thức

BẤT ĐẲNG THỨC –BẤT PHƯƠNG TRÌNH

4

BẤT ĐẲNG THỨC

§BÀI 1.
A–LÝ THUYẾT
1. Định nghĩa: Cho a, b là hai số thực.

Các mệnh đề " a  b ", " a  b ", " a  b ", " a  b " được gọi là những bất đẳng thức.
Chứng minh bất đẳng thức là chứng minh bất đẳng thức đó đúng (mệnh đề đúng)
Với A, B là mệnh đề chứa biến thì " A  B " là mệnh đề chứa biến.
Chứng minh bất đẳng thức A  B (với điều kiện nào đó) nghĩa là chứng minh mệnh đề chứa
biến " A  B " đúng với tất cả các giá trị của biến (thỏa mãn điều kiện đó). Khi nói ta có bất
đẳng thức A  B mà khơng nêu điều kiện đối với các biến thì ta hiểu rằng bất đẳng thức đó
xảy ra với mọi giá trị của biến là số thực.
2. Tính chất :
Tính chất
Tên gọi
Điều kiện
Nội dung
Tính chất bắc cầu
a  b và b  c thì a  c
a b  ac bc
Cộng hai vế của bất đẳng thức với một số
c0

a  b  ac  bc
Nhân hai vế của bất đẳng thức với một số
c0
a  b  ac  bc
Cộng hai bất đẳng thức cùng chiều
a  b và c  d  a  c  b  d
a  0, c  0
Nhân hai bất đẳng thức cùng chiều
a  b và c  d  ac  bd

n



n   và a  0
a0

a  b  a 2 n1  b2 n1

a  b  a 2n  b2n

ab a  b

ab 3 a  3 b

Nâng hai vế của bất đẳng thức lên một lũy
thừa
Khai căn hai vế của một bất đẳng thức

2.1. Ví dụ minh họa:

Ví dụ 1. Cho các số thực a, b, c là số thực. Chứng minh rằng:
a). a  b  c  ab  bc  ca
c). a 2  b 2  c 2  3  2(a  b  c)

b). a 2  b2  1  ab  a  b
d). a 2  b 2  c 2  2(ab  bc  ca )

Lời giải
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Chương IV-Bài 1. Bất Đẳng Thức

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3. Bất đẳng thức về giá trị tuyệt đối.
Điều kiện
Với mọi số thực x .

Nội dung

x  0, x  x, x   x

x  a  a x  a

a0

 x  a
x a  
x  a
a  b  ab  a  b

3.1. Ví dụ minh họa:
Ví dụ 2. Tìm giá trị nhỏ nhất cảu các biểu thức sau
a). A  x  2  x  5 .
b). B  x  3  x  1  x  1  x  3 .

Lời giải
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







Ví dụ 3. Tìm giá trị nhỏ nhất của hàm số y  x  2 1  x  1  x  2 1  x  1 .

Lời giải
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4. Bất đẳng thức giữa trung bình cộng và trung bình nhân (Bất đẳng thức Cauchy)
a). Đối với hai số không âm
ab
Cho a  0, b  0 , ta có
 ab
2
Dấu '  ' xảy ra khi và chỉ khi a  b
Hệ quả :
Hai số dương có tổng khơng đổi thì tích lớn nhất khi hai số đó bằng nhau tức là

 ab
ab  

 2 

2

Hai số dương có tích khơng đổi thì tổng nhỏ nhất khi hai số đó bằng nhau

a  b  2 ab
2

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b). Đối với ba số không âm
Cho a  0, b  0, c  0 , ta có

Chương IV-Bài 1. Bất Đẳng Thức

abc 3
 abc
3

Dấu '  ' xảy ra khi và chỉ khi a  b  c

c). Ví dụ minh họa:
Ví dụ 4. Cho a, b, c là các số thực dương. Chứng minh rằng
1 1 1
9
1 
1 
1

a).  a   b   c    8 .
b).   
.
b 
c 
a
a b c abc


a b c 1 1 1
ab bc ac
c). 2  2  2   
d).
   abc
b c a
a b c
c
a b
Lời giải
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Trung Tâm Luyện Thi Đại Học Amsterdam

Chương IV-Bài 1. Bất Đẳng Thức

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Ví dụ 5. Tìm giá trị nhỏ nhất của các hàm số sau
2
2
a). f  x   x 
với x  1 .
b). f  x   x 
với x  2 .
x 1
x2

Lời giải
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B. CÁC DẠNG TOÁN VÀ PHƯƠNG PHÁP GIẢI.
DẠNG TOÁN 1: SỬ DỤNG ĐỊNH NGHĨA VÀ TÍCH CHẤT CƠ BẢN.
1. Phương pháp.
Để chứng minh bất đẳng thức(BĐT) A  B ta có thể sử dụng các cách sau:
Ta đi chứng minh A  B  0 . Để chứng minh nó ta thường sử dụng các hằng đẳng thức để
phân tích A  B thành tổng hoặc tích của những biểu thức khơng âm.
Xuất phát từ BĐT đúng, biến đổi tương đương về BĐT cần chứng minh.
2. Bài tập minh họa.
Loại 1: Biến đổi tương đương về bất đẳng thức đúng.
Bài tập 1. Cho ba số thực a, b, c . Chứng minh rằng các bất đẳng thức sau

a 2  b2
a). ab 
2

c). 3  a 2  b 2  c 2    a  b  c 

 ab
b). ab  

 2 
2

2

d).  a  b  c   3  ab  bc  ca 
2

Lời giải
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Trung Tâm Luyện Thi Đại Học Amsterdam

Chương IV-Bài 1. Bất Đẳng Thức

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Nhận xét: Các BĐT trên được vận dụng nhiều, và được xem như là "bổ đề" trong chứng minh các
bất đẳng thức khác.
Bài tập 2. Cho năm số thực a, b, c, d , e . Chứng minh rằng

a 2  b 2  c 2  d 2  e 2  a (b  c  d  e) .

Lời giải
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Bài tập 3. Chứng minh rằng
a). a  b  c  ab  bc  ca với a, b, c là các số thực dương.
b). a 2  b2  c 2  3  2  a  b  c  với a, b, c là các số thực.

Lời giải
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1
1
2
.

 2

a  1 b  1 1  ab
Lời giải
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Bài tập 4. Cho ab  1. Chứng minh rằng :

2

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5

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Chương IV-Bài 1. Bất Đẳng Thức

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Nhận xét : Nếu 1  b  1 thì BĐT có chiều ngược lại :

1
1
2
.
 2

a  1 b  1 1  ab
2

Bài tập 5. Cho số thực x . Chứng minh rằng
a). x 4  3  4 x
b). x 4  5  x 2  4 x
c). x12  x 4  1  x9  x
Lời giải

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Bài tập 6. Cho a, b, c là các số thực. Chứng minh rằng
a). a 4  b4  4ab  2  0


c). 3  a

  

 b   ab  4  2  a

b). 2 a 4  1  b2  1  2  ab  1
2

2

2

2

b2  1  b a 2  1



Lời giải
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Chương IV-Bài 1. Bất Đẳng Thức

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Bài tập 7. Cho hai số thực x, y thỏa mãn x  y . Chứng minh rằng;
a). 4  x3  y 3    x  y 

b). x 3  3 x  4  y 3  3 y

3

Lời giải
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Loại 2. Xuất phát từ một BĐT đúng ta biến đổi đến BĐT cần chứng minh
1. Phương pháp.
Đối với loại này thường cho lời giải không được tự nhiên và ta thường sử dụng khi các biến có
những ràng buộc đặc biệt.

Chú ý hai mệnh đề sau thường dùng:
a   ;     a    a     0 *
a, b, c   ;     a    b    c        a    b    c   0 **

2. Bài tập minh họa.
Bài tập 8. Cho a, b, c là độ dài ba cạnh tam giác. Chứng minh rằng :

a 2  b 2  c 2  2(ab  bc  ca ) .
Lời giải
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Chương IV-Bài 1. Bất Đẳng Thức

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Nhận xét : Ở trong bài toán trên ta đã xuất phát từ BĐT đúng đó là tính chất về độ dài ba cạnh
của tam giác. Sau đó vì cần xuất hiện bình phương nên ta nhân hai vế của BĐT với c.
Ngoài ra nếu xuất phát từ BĐT | a  b | c rồi bình phương hai vế ta cũng có được kết quả.

Bài tập 9. Cho tam giác ABC có cạnh a, b, c . Chứng minh rằng nửa chu vi lớn hơn độ dài
mỗi cạnh.
Lời giải
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Bài tập 10. Cho a, b, c  [0;1] . Chứng minh : a 2  b2  c 2  1  a 2b  b 2c  c 2 a
Lời giải
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Bài tập 11. Cho các số thực a, b, c thỏa mãn : a 2  b2  c 2  1 . Chứng minh :
2(1  a  b  c  ab  bc  ca)  abc  0 .
Lời giải
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Chương IV-Bài 1. Bất Đẳng Thức

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Bài tập 12. Chứng minh rằng nếu a  4, b  5, c  6 và a 2  b 2  c 2  90 thì a  b  c  16
Lời giải
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Bài tập 13. Cho ba số a, b, c thuộc  1;1 và không đồng thời bằng không.
Chứng minh rằng

a 4b 2  b 4 c 2  c 4 a 2  3
2
a 2012  b 2012  c 2012

Lời giải
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3. Bài tập luyện tập
Bài 1. Cho a, b, c, d là số dương.. Chứng minh rằng
a ac
a
a). 
với  1 .
b bc
b
a
b
c
b).


2
ab bc ca
a
b
c

d
c). 1 



2
abc bcd cd a d a b
ab
bc
cd
d a
d). 2 



3
a bc bc d c d a d a b
Lời giải
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Chương IV-Bài 1. Bất Đẳng Thức

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Bài 2. Chứng minh các bất đẳng thức sau
a). (ax  by )(bx  ay )  (a  b) 2 xy ( với a, b  0; x, y  R ) .
ca
cb

b).
. với a  b  0; c  ab .
2

2
c a
c2  b2
ab
cb
1 1 2
c).

 4 với a, b, c  0 và  
2a  b 2c  b
a c b
2
2
2
3
d). a (b  c)  b(c  a )  c(a  b)  a  b3  c 3 với a, b, c là ba cạnh của tam giác
Lời giải
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Chương IV-Bài 1. Bất Đẳng Thức

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Bài 3. Cho x  y  z  0 . Chứng minh rằng:
a). xy 3  yz 3  zx 3  xz 3  zy 3  yx 3

b).

x2 y y 2 z z 2 x x2 z y 2 x z 2 y
.






z
x
y
y
z
x

Lời giải
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Bài 4. Cho bốn số dương a, b, c, d . Chứng minh rằng:

1
1 1

a b



1
1 1

c d



1
1
1

ac bd

.

Lời giải
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Bài 5. Cho a, b, c  1;3 và thoả mãn điều kiện a  b  c  6 .
Chứng minh rằng a 2  b2  c 2  14

Lời giải
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Chương IV-Bài 1. Bất Đẳng Thức

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4. Câu hỏi trắc nghiệm.
Câu 1. Trong các khẳng định sau, khẳng định nào sau đây đúng?
a  b
a  b
A. 
B. 
 a  c  b  d.
 a  c  b  d.
c  d
c  d

a  b
a  b  0
C. 
D. 
 a  d  b  c.
 a  c  b  d.
c  d
c  d  0
Lời giải
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Câu 2. Trong các khẳng định sau, khẳng định nào sau đây sai?
a  b
a  b
bc
A. 
a
. B. 
 a  c  b  a. C. a  b  a  c  b  c. D. a  b  c  a  c  b.
2
a  c
a  c

Lời giải

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Câu 3. Trong các khẳng định sau, khẳng định nào đúng?
a  b
a  b
0  a  b
a  b
A. 
 ac  bd . B. 
 ac  bd . C. 
 ac  bd . D. 
  ac   bd .
c  d
c  d

0  c  d
c  d
Lời giải
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Câu 4. Trong các khẳng định sau, khẳng định nào sau đây đúng?
A. a  b  ac  bc.

B. a  b  ac  bc.

C. c  a  b  ac  bc.

a  b
D. 
 ac  bc.
c  0

Lời giải
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Chương IV-Bài 1. Bất Đẳng Thức

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Câu 5. Trong các khẳng định sau, khẳng định nào sau đây đúng?
0  a  b
a  b  0
a  b
a  b  0
a b
a b
a b

a d
A. 
D. 
  . B. 
  . C. 
  .
  .
c d
c d
c d
0  c  d
c  d  0
c  d
c  d  0 b c
Lời giải
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Câu 6. Nếu a  2c  b  2c thì bất đẳng thức nào sau đây đúng?
A.  3a   3b.

B. a 2  b 2 .

C. 2a  2b.

D.

1 1

 .
a b

Lời giải
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Câu 7. Nếu a  b  a và b  a  b thì bất đẳng thức nào sau đây đúng?
A. ab  0.
B. b  a.
C. a  b  0.
D. a  0 và b  0.
Lời giải
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Câu 8. Nếu 0  a  1 thì bất đẳng thức nào sau đây đúng?
1

1
A.  a .
B. a  .
C. a  a .
D. a 3  a 2 .
a
a
Lời giải
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Chương IV-Bài 1. Bất Đẳng Thức

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Câu 9. Cho hai số thực dương a, b. Bất đẳng thức nào sau đây đúng?
A.

a2
1
 .
4
a 1 2

ab 1
 .
ab  1 2

B.

C.

a2  1 1
 .
a2  2 2

D. Tất cả đều đúng.

Lời giải
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1 a
1 b
, y
. Mệnh đề nào sau đây đúng?
2
1 a  a
1  b  b2
A. x  y.
B. x  y.
C. x  y.
D. Không so sánh được.
Lời giải
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Câu 10. Cho a, b  0 và x 

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DẠNG TOÁN 2: SỬ DỤNG BẤT ĐẲNG THỨC CAUCHY(côsi) ĐỂ CHỨNG MINH BẤT ĐẲNG THỨC
VÀ TÌM GIÁ TRI LỚN NHẤT, NHỎ NHẤT.
1. Phương pháp.
Một số chú ý khi sử dụng bất đẳng thức côsi:
Khi áp dụng bất đẳng thức cơsi thì các số phải là những số không âm.
BĐT côsi thường được áp dụng khi trong BĐT cần chứng minh có tổng và tích.
Điều kiện xảy ra dấu ‘=’ là các số bằng nhau.
Bất đẳng thức cơsi cịn có hình thức khác thường hay sử dụng
 Đối với hai số: x  y  2 xy;
2

2

( x  y)2
x y 
;
2
2

2

 x y
xy  
 .
 2 

2

a 3  b3  c 3
 abc 
, abc  
 Đối với ba số: abc 
 .
3
3


3

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Chương IV-Bài 1. Bất Đẳng Thức

2. Các ví dụ minh họa.
Loại 1: Vận dụng trực tiếp bất đẳng thức côsi.

 Bài tập 14. Cho a, b là số dương thỏa mãn a 2  b2  2 . Chứng minh rằng
5
 a b  a b 

a).    2  2   4
b).  a  b   16ab 1  a 2 1  b 2 
 b a  b a 
Lời giải
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 Bài tập 15. Cho a, b, c là số dương. Chứng minh rằng
1 
1 
1

a).  a   b   c    8
b). a 2 (1  b 2 )  b 2 (1  c 2 )  c 2 (1  a 2 )  6abc
b 
c 
a

c). (1  a)(1  b)(1  c)  1  3 abc 

3

d). a 2 bc  b 2 ac  c 2 ab  a 3  b3  c 3

Lời giải

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Chương IV-Bài 1. Bất Đẳng Thức

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Bài tập 16. Cho a, b, c, d là số dương. Chứng minh rằng
abcd 4
a).
 abcd
4
a b c d 
b).  3  3  3  3   a  b  b  c   16
a 
b c d
abc
8abc

 4.
c). 3
(a  b)(b  c)(c  a )
abc
Lời giải
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16

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Trung Tâm Luyện Thi Đại Học Amsterdam

Chương IV-Bài 1. Bất Đẳng Thức

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Loại 2: Kĩ thuật tách, thêm bớt, ghép cặp.
1. Phương pháp.
Để chứng minh BĐT ta thường phải biến đổi (nhân chia, thêm, bớt một biểu thức) để tạo
biểu thức có thể giản ước được sau khi áp dụng BĐT cơsi.
Khi gặp BĐT có dạng x  y  z  a  b  c (hoặc xyz  abc ), ta thường đi chứng minh
x  y  2a (hoặc ab  x 2 ), xây dựng các BĐT tương tự rồi cộng(hoặc nhân) vế với vế ta suy
ra điều phải chứng minh.
Khi tách và áp dụng BĐT côsi ta dựa vào việc đảm bảo dấu bằng xảy ra (thường dấu bằng
xảy ra khi các biến bằng nhau hoặc tại biên).

2. Bài tập minh họa.
Bài tập 17. Cho a, b, c là số dương. Chứng minh rằng:
ab bc ac
a b
c 1 1 1
a).
b). 2  2  2   
 

 abc
c
a b
b c
a
a b c
Lời giải
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17

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Trung Tâm Luyện Thi Đại Học Amsterdam

Chương IV-Bài 1. Bất Đẳng Thức

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Bài tập 18. Cho a, b, c dương sao cho a 2  b2  c 2  3 . Chứng minh rằng
a).

a 3b3 b3c3 c3a 3



 3abc
c
a
b

b).

ab bc ca
 
3.
c
a b

Lời giải
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18

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Tel: 0935.660.880


Trung Tâm Luyện Thi Đại Học Amsterdam

Chương IV-Bài 1. Bất Đẳng Thức

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Bài tập 19. Cho a, b, c là số dương thỏa mãn a  b  c  3 . Chứng minh rằng
a). 8  a  b  b  c  c  a    3  a  3  b  3  c 
b).  3  2a  3  2b  3  2c   abc

Lời giải
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Bài tập 20. Cho a, b, c là số dương. Chứng minh rằng

a2
b2
c2
abc



.
bc ca ab
2

Lời giải
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19

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Trung Tâm Luyện Thi Đại Học Amsterdam

Chương IV-Bài 1. Bất Đẳng Thức

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a2
bc


và đánh giá như trên là vì những lí do sau:
bc
4
Thứ nhất là ta cần làm mất mẫu số ở các đại lượng vế trái (vì vế phải khơng có phân số), chẳng
a2
hạn đại lượng
khi đó ta sẽ áp dụng BĐT cơsi cho đại lượng đó với một đại lượng chứa b  c
bc
.
Thứ hai là ta cần lưu ý tới điều kiện xảy ra đẳng thức ở BĐT cơsi là khi hai số đó bằng nhau. Ta
a2
a
 và b  c  2a do đó ta ghép như trên.
dự đoán dấu bằng xảy ra khi a  b  c khi đó
bc 2
Bài tập 21. Cho a, b, c là số dương thỏa mãn a  b  c  3 . Chứng minh rằng:
Lưu ý : Việc ta ghép

a).

a
b
c
3 2



2
b 1

c 1
a 1

b).

a3
b3
c3
3



b3
c3
a3 2

Lời giải
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20

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Trung Tâm Luyện Thi Đại Học Amsterdam

Chương IV-Bài 1. Bất Đẳng Thức

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Bài tập 22. Cho a, b, c là số dương thỏa mãn abc  1 .
1 1 1
Chứng minh rằng 2  2  2  3  2  a  b  c  .
a b c
Lời giải
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Bài tập 23. Tìm giá trị nhỏ nhất của biểu thức
a).

 x  1
f ( x) 
x2

c). h  x   x 

2

với x  2

3
với x  2
x

b). g ( x)  2 x 
d). k  x   2 x 

1

 x  1

2

với x  1

1
1
với 0  x  .
2
x
2

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21

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Trung Tâm Luyện Thi Đại Học Amsterdam

Chương IV-Bài 1. Bất Đẳng Thức

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Bài tập 24. Cho x, y , z dương. Chứng minh rằng

2 y
2 x
2 z
1 1 1
 3 2 3 2 2 2 2.
3
2
x y
y z
z x
x
y

z

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Bài tập 25. Cho các số dương x, y, z thỏa mãn xyz  1 . Chứng minh rằng:

1  x3  y 3
1  y3  z3
1  z 3  x3


3 3
xy
yz
zx

Lời giải
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22

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Chương IV-Bài 1. Bất Đẳng Thức

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3. Câu hỏi trắc nghiệm.

2
với x  1.
x 1
C. m  1  2.

Câu 11. Tìm giá trị nhỏ nhất m của hàm số f  x   x 
A. m  1  2 2.

B. m  1  2 2.

D. m  1  2.

Lời giải
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Câu 12. Tìm giá trị nhỏ nhất m của hàm số f  x  
A. m  2.

B. m  1.

x2  5
x2  4

.

5
C. m  .
2

D. Không tồn tại m.

Lời giải
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Câu 13. Tìm giá trị nhỏ nhất m của hàm số f  x  
A. m  0.
23

B. m  1.

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x2  2x  2
với x  1.
x 1
C. m  2.
D. m  2.

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Câu 14. Tìm giá trị nhỏ nhất m của hàm số f  x  
A. m  4.

B. m  18.

 x  2  x  8
x
C. m  16.

với x  0.

D. m  6.

Lời giải
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Câu 15. Tìm giá trị nhỏ nhất m của hàm số f  x  
A. m  2.

B. m  4.

4
x
với 1  x  0.

x 1 x

C. m  6.

D. m  8.

Lời giải
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Câu 16. Tìm giá trị nhỏ nhất m của hàm số f  x  
A. m  2.

B. m  4.

1
1
với 0  x  1.

x 1 x
C. m  8.

D. m  16.

Lời giải
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24

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Trung Tâm Luyện Thi Đại Học Amsterdam

Chương IV-Bài 1. Bất Đẳng Thức

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Câu 17. Tìm giá trị nhỏ nhất m của hàm số f  x  

1
A. m  .
2

7
B. m  .
2

x 2  32
với x  2.
4  x  2
C. m  4.

D. m  8.

Lời giải
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Câu 18. Tìm giá trị nhỏ nhất m của hàm số f  x  
A. m  2.

B. m  4.

2 x3  4
với x  0.
x
C. m  6.

D. m  10.

Lời giải
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Câu 19. Tìm giá trị nhỏ nhất m của hàm số f  x  
A. m  4.

B. m  6.

x4  3
với x  0.
x
13
C. m  .
2

D. m 

19
.
2

Lời giải
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25

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